[Math] Is it true that if $x_n$ converges and $y_n$ is bounded, then $x_ny_n$ converges

Question

Is it true that if $x_n$ converges and $y_n$ is bounded, then $x_ny_n$ converges?

$x_n$ is said to be bounded if and only if it is bounded both above and below.

I believe this to be false. My attempt at a counterexample:

Let $x_n=(-1)^n(1-\frac{1}{n})$ and $y_n=(-1)^{3n}+2$

Then, $x_n$ converges to $1$ and $y_n$ will be bounded above at $3$ and bounded below at $1$

Thus, $x_ny_n$ will produce the sequence: $0,\frac{3}{2},-\frac{2}{3},\frac{9}{4}…$ that does not converge.

Is this an appropriate counterexample?

Answer 1

Following GitGud comment: A general set of examples will consist of $x_n\equiv c$ where $c\neq 0 $ is a constant and $y_n$ as any bounded non-convergent sequence